Parabolic Profile in Heat-Conduction Problems. 2. Semi-Bounded Space with a Time-Varying Surface Temperature

Abstract

Some approaches to the optimization of the exponent n in the parabolic temperature profile for a semi-bounded space at a boundary condition with a time-varying surface temperature are considered. It is shown that solutions obtained using the scheme of T. Myers with minimization of the Langford norm are not even close to the optimum ones and that the minimization scheme based on the new error norm estimating the modulus of a temperature deviation is much more efficient. New hybrid integral schemes convenient to use are proposed. Parabolic solutions have been obtained on the basis of a number of integral relations: the temperature-momentum integrals of single and double modifications and the temperature-function integral. On the basis of these solutions in combination with known integral relations, different variations of the hybrid integral method have been developed. It was established that the highest approximation accuracy is provided by the hybrid scheme combining the temperature-momentum integrals of single and double modifications. It is shown that the hybrid integral schemes proposed are much more accurate compared to the known integral methods, including those realized using the scheme with minimization of the Langford norm. In particular, for a semi-bounded space with a surface of constant temperature, the hybrid integral method variant combining the temperature-momentum integral of single modification and the refined integral method makes it possible to obtain a simple cubic parabola representing a substantially better approximation solution compared to the temperature profile obtained on the basis of the refined integral method with optimization of the exponent n by the scheme of T. Myers.